The phrase "give eyeteeth for" means to want something desperately or to sacrifice a lot for it. Antonyms for this phrase can include not being interested or not caring at all, being indifferent or uninterested, despising or disliking the thing in question, or simply being content with what one already has. Being unenthused, unambitious, dispassionate, blase, or cool towards something are also possible antonyms. This phrase is typically used when talking about something very desirable or precious, and antonyms can range from being dismissive and apathetic to completely adverse or hostile. Ultimately, the antonym for "give eyeteeth for" depends on the context and the speaker's attitude towards the thing in question.